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Finite element approximation of steady flows of generalized Newtonian fluids with concentration-dependent power-law index


Authors: Seungchan Ko and Endre Süli
Journal: Math. Comp. 88 (2019), 1061-1090
MSC (2010): Primary 65N30; Secondary 74S05, 76A05
DOI: https://doi.org/10.1090/mcom/3379
Published electronically: September 10, 2018
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Abstract: We consider a system of nonlinear partial differential equations, modeling the motion of a viscous incompressible chemically reacting generalized Newtonian fluid in three space dimensions. The governing system consists of a steady convection-diffusion equation, for the concentration, and a generalized steady power-law-type fluid flow model, for the velocity and the pressure of the fluid, where the viscosity depends on both the shear-rate and the concentration through a concentration-dependent power-law index. The aim of the paper is to perform the mathematical analysis of a finite element approximation of this model. We consider a regularization of the model by introducing an additional term in the momentum equation and construct a finite element approximation of the regularized system. First, the convergence of the finite element method to a weak solution of the regularized model is shown, and we then prove that weak solutions of the regularized problem converge to a weak solution of the original problem.


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Additional Information

Seungchan Ko
Affiliation: Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom
Email: ksm0385@gmail.com

Endre Süli
Affiliation: Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom
Email: endre.suli@maths.ox.ac.uk

DOI: https://doi.org/10.1090/mcom/3379
Keywords: Non-Newtonian fluid, variable-exponent spaces, synovial fluid, finite element method
Received by editor(s): September 1, 2017
Received by editor(s) in revised form: April 3, 2018
Published electronically: September 10, 2018
Additional Notes: The first author’s work was supported by the UK Engineering and Physical Sciences Research Council [EP/L015811/1].
Article copyright: © Copyright 2018 American Mathematical Society